When a ball is freely fallen from a given height,it bounces to $80\%$ of its original height. What fraction of its mechanical energy is lost in each bounce?

$A$ bullet of mass $0.01 \, kg$ travelling at a speed of $500 \, m/s$ strikes a block of mass $2 \, kg$ which is suspended by a string of length $5 \, m$. The center of gravity of the block is found to rise a vertical distance of $0.1 \, m$. What is the speed of the bullet after it emerges from the block?

$A$ simple pendulum of length $1 \,m$ has a wooden bob of mass $M = 1 \,kg$. It is struck by a bullet of mass $m = 10^{-2} \,kg$ moving with a speed of $u = 2 \times 10^2 \,m/s$. The bullet gets embedded into the bob. The height to which the bob rises before swinging back is (use $g = 10 \,m/s^2$): (in $\,m$)

Statement $(I)$: The slope of the kinetic energy-displacement curve of a body in motion is directly proportional to its acceleration.
Statement $(II)$: From a height of $15 \ m$, a ball is projected vertically upwards with a velocity of $30 \ m/s$. If the ball rises to the same height after hitting the ground, the loss of its energy on hitting the ground is $30 \%$.
Statement $(III)$: The velocity acquired by a body of mass '$m$' after travelling a fixed distance from rest under the action of a constant force is directly proportional to mass '$m$'.
Which of the following is correct?

Two balls,having linear momenta $\vec{p}_1 = p \hat{i}$ and $\vec{p}_2 = -p \hat{i}$,undergo a collision in free space. There is no external force acting on the balls. Let $\vec{p}_1^{\prime}$ and $\vec{p}_2^{\prime}$ be their final momenta. Which of the following option$(s)$ is (are) $NOT ALLOWED$ for any non-zero value of $p, a_1, a_2, b_1, b_2, c_1$ and $c_2$?
$(A)$ $\vec{p}_1^{\prime} = a_1 \hat{i} + b_1 \hat{j} + c_1 \hat{k}$,$\vec{p}_2^{\prime} = a_2 \hat{i} + b_2 \hat{j}$
$(B)$ $\vec{p}_1^{\prime} = c_1 \hat{k}$,$\vec{p}_2^{\prime} = c_2 \hat{k}$
$(C)$ $\vec{p}_1^{\prime} = a_1 \hat{i} + b_1 \hat{j} + c_1 \hat{k}$,$\vec{p}_2^{\prime} = a_2 \hat{i} + b_2 \hat{j} - c_1 \hat{k}$
$(D)$ $\vec{p}_1^{\prime} = a_1 \hat{i} + b_1 \hat{j}$,$\vec{p}_2^{\prime} = a_2 \hat{i} + b_1 \hat{j}$

$A$ sphere of mass $m$ moving with a constant velocity $u$ hits another stationary sphere of the same mass. If $e$ is the coefficient of restitution,then the ratio of the velocity of the two spheres after the collision will be